Question

Let y be a continuous uniform random variable, Y - Gumbel(B).for ß>0. That is, Y has cumulative density function PIY <y)=Fly)=e for YER. Showing all of your working, find the probability density function of Show that the inverse of the cumulative density function is given by F (y)=u-Bin(–In(y)). for YER. Given realisations {u,, uz,...,Ug} = {0.710,0.119,0.358,0.883,0.504} of a U[0, 1] variable, generate five realisations {y, Y2,..., Ys} of Y-Gumbel(5, 10). Clearly explain your method and any calculations required.

Answer 1

- liefly) - äeßt getr, so fly) - 8 d Fly) et-e ( By chain rute) neue e so fly YER Ely - e need to find invase of Fly) write y in terms og Flyy

Take log both sides to the base e dog Fly) = ' -e - log Fly) = e log Fly) e log (log ) - {y- Blog Ilog (11) 44-4% y = M - $ log (log ) y = M- plog 1-log Fly)) (as log (1) t = -logx) of variable so writing gas Fly) an and fly, asy Le change of name, f (y) = M- plogl-logy) QER I log and tu are to samme is to when log is to the base t

No. Date : 1 1 u Need to generate 5 realisation from Gumbel ( 5 10) covesponding I to landom uniform 40.40 0.119, 0.356 -0.88 3,0504? so gi = F (Ui) Here "=5 .ß= 10, El now for Ui =0.710 yi = 5-10x (log (- log 0-710)) 35-10% log (-beck 0.3421) = 5 - 10 x log 10:342) 35-10xEL. 0729)) -5+0-7204 = 15.7294 similaily for other vi for U; = 0.119, it is equal to - 2.5547 for Ui = 0.358 Yi = 4:73 tou ; - 0.883 Hi = 25.84 for i = 0.504 gis 8.78 4 calculation similar to above obtain that ang any calculator)

No. Date : 1 1 so realisation of y, ys} are 115.73; -2.55, 4.73, 25.84, 8.787,